Subgroups — C₂₃×#20459 #19168085

Order 5888.

994 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#430	C₂	1	2	yes	maximal
2	C₂₃×#2873	C₂	1	2	yes	maximal
3	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1778	C₂	1	4	yes	maximal, complemented
5	C₄×C₂₃×#108	C₂	1	4	yes	maximal, complemented
6	C₂×C₂₃×#341	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₂₃	C₂²	1	4	yes	—
8	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
11	C₂₃×#430	C₄	1	16	yes	complemented
12	C₂×C₂₃×#108	C₂²	1	8	yes	—
13	C₂₃×#405	—	2	8	no	—
14	C₂₃×#405	C₄	1	8	yes	—
15	C₄×M₁₆×C₂₃	C₂²	1	8	yes	—
16	C₂×C₂₃×#97	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#341	C₂²	1	4	yes	—
19	C₂₃×#341	C₂²	1	4	yes	—
20	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₂₃	—	2	4	no	—
22	C₂×M₁₆×C₂₃	—	2	8	no	—
23	C₂×M₁₆×C₂₃	C₂×C₄	1	16	yes	—
24	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
25	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
26	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
27	C₂²×C₈×C₂₃	—	2	4	no	—
28	C₂²×C₈×C₂₃	—	2	4	no	—
29	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
33	C₂₃×#117	C₂×C₄	1	4	yes	—
34	C₂₃×#117	D₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₂₃	Q₈	1	2	yes	—
36	C₂×C₂²⋊C₄×C₂₃	Q₈	1	2	yes	—
37	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
38	C₂₃×#108	C₂×C₄	1	32	yes	—
39	C₂₃×#97	—	2	8	no	—
40	C₂₃×#97	C₂×C₄	1	8	yes	—
41	C₄×C₈×C₂₃	—	2	8	no	—
42	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
43	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂×Q₈	1	4	yes	—
48	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	#41	1	4	yes	—
50	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
51	M₁₆×C₂₃	—	2	16	no	—
52	M₁₆×C₂₃	—	2	8	no	—
53	M₁₆×C₂₃	C₄²	1	16	yes	—
54	M₁₆×C₂₃	C₂²×C₄	1	16	yes	—
55	C₂×C₈×C₂₃	—	2	8	no	—
56	C₂×C₈×C₂₃	—	2	16	no	—
57	C₂×C₈×C₂₃	—	2	8	no	—
58	C₂×C₈×C₂₃	—	2	2	no	—
59	C₂×C₈×C₂₃	—	2	2	no	—
60	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
63	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
64	C₄⋊C₄×C₂₃	C₄⋊C₄	1	8	yes	—
65	C₂²⋊C₄×C₂₃	C₄⋊C₄	1	8	yes	—
66	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
67	C₄²×C₂₃	C₂×D₄	1	4	yes	—
68	#20459	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₂₃	—	4	2	no	—
70	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
71	C₂³×C₂₃	C₄×Q₈	1	4	yes	—
72	C₂³×C₂₃	C₂²×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₂₃	—	4	4	no	—
74	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
75	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
76	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₂₃	#117	1	4	yes	—
78	C₂×C₄×C₂₃	C₄×D₄	1	8	yes	—
79	C₂×C₄×C₂₃	C₄×Q₈	1	4	yes	—
80	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
82	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
84	C₈×C₂₃	—	2	2	no	—
85	C₈×C₂₃	—	2	2	no	—
86	C₈×C₂₃	—	2	16	no	—
87	C₈×C₂₃	C₂×C₄²	1	8	yes	—
88	C₂×#430	C₂×C₂₃	1	2	yes	—
89	#2873	C₂×C₂₃	1	2	yes	—
90	C₂×#405	C₂×C₂₃	1	1	yes	characteristic
91	#1778	C₂×C₂₃	1	4	yes	complemented
92	C₄×#108	C₂×C₂₃	1	4	yes	complemented
93	C₂×#341	C₂×C₂₃	1	2	yes	—
94	C₂²×C₂₃	—	2	2	no	—
95	C₂²×C₂₃	—	2	4	no	—
96	C₂²×C₂₃	—	4	4	no	—
97	C₂²×C₂₃	C₄×C₄⋊C₄	1	4	yes	—
98	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₂₃	C₂×C₄×Q₈	1	2	yes	—
100	C₄×C₂₃	—	2	8	no	—
101	C₄×C₂₃	C₄×C₄⋊C₄	1	4	yes	—
102	C₄×C₂₃	C₂²×C₄²	1	1	yes	characteristic
103	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
104	C₄×C₂₃	C₂×C₄×D₄	1	2	yes	—
105	C₂²×M₁₆	C₂²×C₂₃	1	4	yes	—
106	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
107	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
108	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
109	#430	C₄×C₂₃	1	16	yes	complemented
110	C₂×#108	C₂²×C₂₃	1	8	yes	—
111	#405	—	2	8	no	—
112	#405	C₄×C₂₃	1	8	yes	—
113	C₄×M₁₆	C₂²×C₂₃	1	8	yes	—
114	C₂×#97	C₂²×C₂₃	1	2	yes	—
115	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
116	#341	C₂²×C₂₃	1	4	yes	—
117	#341	C₂²×C₂₃	1	4	yes	—
118	C₂×C₂₃	—	2	2	no	—
119	C₂×C₂₃	—	2	4	no	—
120	C₂×C₂₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₂₃	#2916	1	2	yes	—
122	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
123	C₂×M₁₆	—	2	4	no	—
124	C₂×M₁₆	—	2	8	no	—
125	C₂×M₁₆	C₂×C₄×C₂₃	1	16	yes	—
126	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
127	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
128	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
129	C₂²×C₈	—	2	4	no	—
130	C₂²×C₈	—	2	4	no	—
131	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
132	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
133	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
134	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
135	#117	C₂×C₄×C₂₃	1	4	yes	—
136	#117	D₄×C₂₃	1	4	yes	—
137	C₂×C₄⋊C₄	Q₈×C₂₃	1	2	yes	—
138	C₂×C₂²⋊C₄	Q₈×C₂₃	1	2	yes	—
139	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
140	#108	C₂×C₄×C₂₃	1	32	yes	—
141	#97	—	2	8	no	—
142	#97	C₂×C₄×C₂₃	1	8	yes	—
143	C₄×C₈	—	2	8	no	—
144	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
145	C₂₃	#20459	1	1	yes	characteristic, direct product factor, Sylow
146	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
147	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
148	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
149	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
150	C₂²×C₄	C₂×Q₈×C₂₃	1	4	yes	—
151	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
152	C₂²×C₄	#41×C₂₃	1	4	yes	—
153	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
154	M₁₆	—	2	16	no	—
155	M₁₆	—	2	8	no	—
156	M₁₆	C₄²×C₂₃	1	16	yes	—
157	M₁₆	C₂²×C₄×C₂₃	1	16	yes	—
158	C₂×C₈	—	2	8	no	—
159	C₂×C₈	—	2	16	no	—
160	C₂×C₈	—	2	8	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
164	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
165	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
166	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
167	C₄⋊C₄	C₄⋊C₄×C₂₃	1	8	yes	—
168	C₂²⋊C₄	C₄⋊C₄×C₂₃	1	8	yes	—
169	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
170	C₄²	C₂×D₄×C₂₃	1	4	yes	—
171	C₂³	—	4	2	no	—
172	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
173	C₂³	C₄×Q₈×C₂₃	1	4	yes	—
174	C₂³	C₂²×Q₈×C₂₃	1	1	yes	characteristic
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
177	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
178	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
179	C₂×C₄	C₂₃×#117	1	4	yes	—
180	C₂×C₄	C₄×D₄×C₂₃	1	8	yes	—
181	C₂×C₄	C₄×Q₈×C₂₃	1	4	yes	—
182	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
184	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
185	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
186	C₈	—	2	2	no	—
187	C₈	—	2	2	no	—
188	C₈	—	2	16	no	—
189	C₈	C₂×C₄²×C₂₃	1	8	yes	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	4	4	no	—
193	C₂²	C₄×C₄⋊C₄×C₂₃	1	4	yes	—
194	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
195	C₂²	C₂×C₄×Q₈×C₂₃	1	2	yes	—
196	C₄	—	2	8	no	—
197	C₄	C₄×C₄⋊C₄×C₂₃	1	4	yes	—
198	C₄	C₂²×C₄²×C₂₃	1	1	yes	characteristic
199	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
200	C₄	C₂×C₄×D₄×C₂₃	1	2	yes	—
201	C₂	—	2	4	no	—
202	C₂	—	2	2	no	—
203	C₂	C₂×C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
204	C₂	C₂₃×#2916	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₂₃	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₂₃×#1778	C₂	semidirect
5. C₄×C₂₃×#108	C₂	semidirect
11. C₂₃×#430	C₄	semidirect
68. #20459	C₂₃	direct (N × H)
91. #1778	C₂×C₂₃	semidirect
92. C₄×#108	C₂×C₂₃	semidirect
109. #430	C₄×C₂₃	semidirect
145. C₂₃	#20459	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#430	2
C₂₃×#2873	2
C₂×C₂₃×#405	2
C₂₃×#1778	2
C₄×C₂₃×#108	2
C₂×C₂₃×#341	2
#20459	23

Subgroups — C23×#20459 #19168085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20459 #19168085